1. Assume (X, o) and (Y,on X x Y asare groups. Let X × Y = {(x, y) |x € X, y € Y} and define the operation *(x1, Y1) * (x2, Y2) = (x1 0 x2, Y1 • Y2)for (x1, y1), (x2,Y2) E X × Y. Show that (X x Y, *) is a group.

The given question is related to group theory. Given: X , ∘ and Y , ∙ are groups. Let X × Y = x,y |…

Answered: Assume (X,o) and (Y, on X x Y as are…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Let A and B be positive operators on H. It is clear that A + B is a positive operator. In general, the composition operator AB may not be a positive operator. For example, let H = K² and A(x(1), x(2)) = (x(1) + x(2), x(1) + 2x(2)), B(x(1), x(2)) = (x(1) + x(2), x(1) + x(2)). Then AB(x(1), x(2)) = (2x(1) + 2x(2), 3x(1) + 3x(2)) Ox= for all (x(1), x(2)) = K². Note that A and B are positive operators. But AB is not a positive operator, since it is not self-adjoint and if x = (-4, 3), then (AB(x), x) = -1. One can show that every positive operator A has a unique positive square root C which commutes with 3 BE BL(H) whenever A commutes with B. One can deduce that if A and B are positive operators and AB = BA, then AB is a positive operator. Request explain 43 how? 00
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Assume (X,o) and (Y, on X x Y as are groups. Let X × Y = {(x, y) | æ € X, y E Y} and define the operation * (x1,Y1) * (x2, Y2) = (x1 © x2, Y1 • Y2) for (a1, Y1), (x2, Y2) E X × Y. Show that (X x Y, *) is a group.